Introduction to Differentiable Manifolds - 57430 - MATH 536 - 002

Lecture Time MW 4:00 pm - 5:15 pm, Location SMLC 352

 

                                                                 

Instructor: Dimiter Vassilev     Office :  SMLC, Office 326  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Please note the following information. This page will not be updated beyond January 30, 2026. You should join the class MS Team and follow the information there. You need to be registered for the course with a @unm.edu email. Any other email will disable features of Microsoft Teams. I will send you by email a link to join the class team; please follow it in order to request access. You can find the link to join the team in Canvas as well.

 

Office Hours: Office Hours: MW 1:30pm - 2:30pm. Feel free to stop-by anytime when you have a quick question.

 

Midterm Exam: Wednesday, March 25. Final exam: Monday, May 11 5:30–7:30 p.m. (usual room). Please double check the official Final Examination Schedule.

Examinations will take place in the same rooms as class meetings unless otherwise announced by the instructor. A change in the final exam day/time must be approved by the instructor's College Dean. Lab exams may be given during the week preceding finals week or during the time period listed below during finals week. It is the student's responsibility to inform their instructors before Friday, April 10, 2026, if they have conflicts with this exam schedule.

 

Textbook:

Other suggested textbook:

Please note the following guidelines for the course:

 

Description: Concept of a manifold, differential structures, vector bundles, tangent and cotangent bundles, embedding, immersions and submersions, transversality, Stokes' theorem. Prerequisite: Math 511. Let me know if you need a waiver or an override.


Please note the following guidelines for the course:

Attendance: Attendance at UNM is mandatory, see policy.

Grades: The final grade will be determined by homework (25%), one midterm (25%) and a final exam (50%). The Final Exam score will replace the midterm scores that are lower than the Final Exam score.  There might be some opportunities for bonus points which will only be announced in class. Although a small curve might be used, 90% , 80% or 70% of the possible maximum points guarantees at least a grade in the ranges of A(A-, A, A+), B (B-, B, B+) or C (C-,C, C+), respectively.

Missed Exams:  Make-up exams can be arranged for exams missed with a VALID excuse (illness, family emergency, active participation in scholarly or athletic activities), and ONLY if prior notice is given unless there was an emergency.

Accommodation Statement: In accordance with University Policy 2310 and the Americans with Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. Accessibility Resources Center (Mesa Vista Hall 2021, 277-3506) provides academic support to students who have disabilities. If you think you need alternative accessible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner.

 

Homework:   Homework is due every Wednesday at the beginning of the class. I encourage you to work on the homework with your classmates, but you are required to write up your own solutions in your own words. To help the grader, please write your solutions neatly using correct grammar and mathematical notation (no points will be given for work that the reader cannot follow).   The ten best homework grades will be used in computing the homework score. Please do not turn-in late homework! The syllabus also lists recommended homework problems.  These are NOT to be handed in. Work as many as it takes for you to understand the material.  You should see me as early and as often as necessary if you are having difficulties with the homework problems .

 

 Covid Policies

 

 The Schedule and Topics will be posted on the class MS Teams  (advanced postings could change) . Below is a typical content for the course.

 

Topics

Topological and smooth manifolds - definitions and examples, paracompactness, partition of unity, smooth structures defined by an atlas.

Continue smooth structures defined by an atlas - examples. Smooth Maps.  Manifolds with boundary. Lie groups. Locally trivial fibrations. Smooth covering maps (HW Pbm).

Bundle maps and morphisms. Vector bundles. Examples. Diffeomorphisms, smooth partitions of unity.

Tangent vectors and the tangent space at a point. The tangent bundle. The rank theorem; immersions and submersions.

Embeddings.  Submanifolds- regular level set, uniqueness of smooth structure; O(n), SL(n). Tangent space to a submanifold. Defining function of the boundary and regular domains. The Whitney embedding theorem. Transversality. Fiber product theorem for transversal maps.

Lie groups and algebras.  Group actions and the quotient manifold theorem. Free and proper group actions and the quotient manifold M/G. Principle fiber bundles.  Homogeneous spaces.

Vector bundles. Tautological line bundle over the projective space. Constructing new bundles from old. Integrable/ non-integrable sub-bundles of the tangent bundle.

The tensor bundle and tensor fields, characterization of (0,q)-tensors. Straightening of a vector field near a regular point.

Frobenius' theorem. Integral curves and the flow of a vector field. Complete vector fields, 1-parameter group of diffeomorphisms, infinitesimal generator.

The Lie derivative of a vector field- commuting flows. Derivations of the tensor algebra. The Lie derivative as a derivation of the tensor algebra. Derivations and skew-derivations of forms, the exterior derivative.

The interior multiplication. Cartan's formula. Closed and exact forms, de Rham cohomology and homotopies. Frobenius' theorem in the language of forms (differential ideals).

Integration of differential forms and Stokes' theorem.

Mayer-Vietoris sequence- cohomology of spheres. The degree of a map, index of a vector field and Hopf's theorem.